# Numerical Anisotropy in Finite Differencing

## TL;DR

This paper reviews numerical anisotropy in finite difference methods for wave equations, highlighting how multi-dimensional discretization causes directional errors and discussing optimization strategies to mitigate these effects.

## Contribution

It provides a comprehensive overview of numerical anisotropy in finite difference schemes and discusses recent optimization approaches to reduce this error in multi-dimensional wave simulations.

## Key findings

- Numerical anisotropy causes directional wave speed errors.
- Optimization techniques can significantly reduce anisotropy.
- Review of key studies on finite difference scheme improvements.

## Abstract

Numerical solutions to hyperbolic partial differential equations, involving wave propagations in one direction, are subject to several specific errors, such as numerical dispersion, dissipation or aliasing. In multi-dimensions, where the waves propagate in all directions, there is an additional specific error resulting from the discretization of spatial derivatives along grid lines. Specifically, waves or wave packets in multi-dimensions propagate at different phase or group velocities, respectively, along different directions. A commonly used term for the aforementioned multidimensional discretization error is the numerical anisotropy or isotropy error. In this review, the numerical anisotropy is briefly described in the context of the wave equation in multi-dimensions. Then, several important studies that were focused on optimizations of finite difference schemes with the objective of reducing the numerical anisotropy are discussed.